The Mathematical Continuum - Finding the living Infinite
‘‘There are two labyrinths of the human mind: one concerns the composition of the continuum, and the other the nature of freedom, and both spring from the same source—the infinite’. —G.W. Leibniz
Brouwer describes his understanding of the problems of the continuum as follows:
“In the following chapters we shall go further into the basic intuition of mathematics (and of every intellectual activity) as the substratum, divested of all quality, of any perception of change, a unity of continuity and discreteness, a possibility of thinking together several entities, connected by a 'between', which is never exhausted by the insertion of new entities. Since continuity and discreteness occur as inseparable complements, both having equal rights and being equally clear, it is impossible to avoid one of them as a primitive entity, trying to construe it from the other one, the latter being put forward as self-sufficient; in fact it is impossible to consider it as self-sufficient. Having recognised that the intuition of continuity, of 'fluidity', is as primitive as that of several things conceived as forming together a unit, the latter being at the basis of every mathematical construction, we are able to state properties of the continuum as a 'matrix of points to be thought of as a whole'. First of all, there is neither a first nor a last point…” ---L. E. J. Brouwer, Writings vol 1.
What is the notion of the mathematical continuum?
The continuum is “an infinite collection whose elements are themselves infinite sets, each of whose elements in turn is an infinite sequence.” Cantor’s set theory makes this concept mathematically rigorous. (Shapiro, 'Oxford Philosophy of Mathematics' pdf p 321)
Phemister says of Leibniz p 116 - “Leibniz’s solution [ is ] that the continuum is composed of indivisible living corporeal substances." These are what Leibniz calls the Monads.
Deleuze describes the connections between such a virtual consciousness of flux and fold and the work of Leibniz:
“En même temps que l'objet devient objectile, le sujet devient superjet. Entre la variation et le point de vue il y a un rapport nécessaire : non pas simplement en raison de la variété des points de vue... mais en premier lieu parce que tout point de vue est point de vue sur une variation. Ce n'est pas le point de vue qui varie avec le sujet, du moins en premier lieu ; il est au contraire la condition sous laquelle un éventuel sujet saisit une variation (métamorphose), ou quelque chose = x ( anamorphose). Le perspectivisme chez Leibniz, et aussi chez Nietzsche, chez William et chez Henry James, chez Whitehead, est bien un relativisme, mais ce n'est pas le relativisme qu'on croit. Ce n'est pas une variation de la vérité d'après le sujet, mais la condition sous laquelle apparaît au sujet la vérité d'une variation. C'est l'idée même de la perspective baroque.”
--- Deleuze, Le Pli et Le Baroque p 27
The essential idea that we uncover and merge with as we consider Deleuze’s comments are as follows. The flowing of folding-object streams (Newton calls these ‘fluents’ in his calculus) melt together with the subject who becomes more than himself in a super-jection. Continual and infinite metamorphosis of infinite curves and flexing on the plane do not just produce an infinite variety, but this continual metamorphosis is the condition on which the subject himself is created. The fact that the spatial point of view changes as the perceiver moves around in space does not exhaust the intuitive vistas that are available to the awakened self-conscious I. The point of view of the subject in baroque geometry and architecture does not merely show relativism, but shows that the infinite becoming of the other at the flexible vanishing point (the folding virtual point) is also the differential origin of any subjectivity. Fluid variation forms the root of the subject.
This is a fine transition zone, where the self-conscious I, that has been trained by the living soul of Anthroposophy, can now perceive what is inside the house and what is outside the window, by becoming a coherent, balanced, larger sense of Self - a multiplicity, coordinated by a Spirit-Self outside of space, time, and personality. This spiritualization of Deleuze’s work is necessary if one is to grasp the spiritual life at the bottom of living or etheric-virtual mathematics. Etheric and spiritual love form the personality and karma forms the vehicles in which his motion takes place.
The opening and loving source of spiritual life has no bottom or end. When I am self-consciously united and merged with this spiritual light, love, and self-perceiving rigorous wakefulness in pure thinking-willing, I can see where the tendencies to infinitely divide every plane come from. This tendency comes from the desire to uncover more of the infinite love that powers all desire and all of my self - to learn from the acts of spiritual beings in the depth of my own spirit. It should be recognized that the representing or willed picturing that projects a web or organization of the dead part of the personal etheric body, imagines that this infinite life can be infinitely subdivided in the sense of cutting a plane, or inserting more space in between two spaces. But this is a concept that controls the parameters of the plane of representation, it is not the pure experience of intuitive willed-thinking, living inside the pure spiritual act of meaning-life. But this concept also controls and organizes the pure flow and chaos of momentary sense perceptions and feelings. The following must be understood quite clearly: when I cut a block of cheese into two large parts, and then subdivide this cheese into smaller bits, and then subdivide these into 50 tiny pieces, let us say, the number of ‘pieces’ is not what is perceived. We represent the number of pieces on the plane of representation which we create unconsciously. In the pure flow of colors and the sense of weight and perspective and changing shades, there are no parts.
The pure flux of sensation and pure percept flow can never be divided. Only consciousness divides, and it does this on the plane of representation or the plane of mathematics. So we actually have two flowing wholes. On the side of pure perception, the pure flux of world-formations. On the other side, the infinite gift of the archetype unveiled in the seed of the intuited pure concept or virtual meaning-light-archetype. It is only in between, in this thin plane where our normal consciousness is constructed by the ahrimanic and luciferic powers, that we find the tendency to want to divide ‘matter’ infinitely into electro-magnetic components in a materialistic way (Ahriman), or imagine ecstatically the numerical infinity that bypasses all human life, such as infinite extent of space, infinite galaxies, never-ending numerical sequences, and so on (Lucifer). That does not mean that we should not school our consciousness with utmost rigor on this mathematical plane, but it does mean that we must self-consciously perceive, create, and freely navigate into the spiritual worlds of life and love that are beyond it. This requires the truly down to earth human and loving Christ-Spirit, teaching us in the bottom of our free I.
Thus we find that the human continuum must be a new seedling, planted by the Christ or the eternal source of turning death into life. I can always find new life in between dead space, there is always more spiritual time in between the moments of my personal biography. The garments of my own life, will be spiritualized by the hierarchies when I hand it over. But this exchange can also be started on the earth.
